Infinite Series Volume Calculation
This problem involves finding the sum of an infinite geometric series. The volumes of the boxes form a sequence where each term is half of the previous one.
Identifying Series Parameters
- The first term (largest box volume), a = $20\text{ cc}$.
- The common ratio, r, is the factor by which each volume decreases. Since each box has half the volume of the previous one, r = $1/2$.
Sum of Infinite Geometric Series
The formula for the sum (S) of an infinite geometric series is:
$ S = \frac{a}{1-r} $
This formula is valid only if the absolute value of the common ratio is less than 1 (i.e., $|r| < 1$). In this case, $|1/2| < 1$, so the formula can be applied.
Calculating Total Volume
Substitute the values of a and r into the formula:
$ S = \frac{20}{1 - \frac{1}{2}} $
First, calculate the denominator:
$ 1 - \frac{1}{2} = \frac{1}{2} $
Now, perform the division:
$ S = \frac{20}{\frac{1}{2}} = 20 \times 2 = 40 $
The total volume of all the boxes is $40\text{ cc}$.
Important Questions from Progression (Notes)
- The sum of 16 terms of the series $\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + .....$ is :
- An auditorium has 8 seats in the first row, with every row to follow having 4 more seats than its preceding row. The total capacity is 416. What is the minimum number of rows needed to seat 150 people?
- The $5^{\text{th}}$ and $9^{\text{th}}$ terms of an arithmetic progression are 7 and 13 respectively. What is the $15^{\text{th}}$ term?
- Find the sum of the G.P.:
$5/11, 5/121, 5/1331, 5/14641, ...$ to $n$ terms. Suppose $a_1, a_2,..., a_{300}$ are integers such that $a_{i-1}+ a_i+ a_{i+1} = 2025$ for all $i = 2,3, ..., 299$.
If $a_7 = -5, a_9 = 37$, then the value of $a_{106}$ is
